Linear Equations in Two Variables
In this chapter, we’ll use the geometry of lines to help us solve equations.
Linear equations in two variables If a, b,andr are real numbers (and if a and b are not both equal to 0) then ax + by = r is called a linear equation in two variables. (The “two variables” are the x and the y.) The numbers a and b are called the coe cients of the equation ax+by = r. The number r is called the constant of the equation ax + by = r.
Examples. 10x 3y =5and 2x 4y = 7 are linear equations in two variables.
Solutions of equations A solution of a linear equation in two variables ax+by = r is a specific point in R2 such that when when the x-coordinate of the point is multiplied by a, and the y-coordinate of the point is multiplied by b, and those two numbers are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in two variables.)
Example. Let’s look at the equation 2x 3y =7. Notice that x =5andy =1isapointinR2 that is a solution of this equation because we can let x =5andy =1intheequation2x 3y =7 and then we’d have 2(5) 3(1) = 10 3=7. The point x =8andy =3isalsoasolutionoftheequation2 x 3y =7 since 2(8) 3(3) = 16 9=7. The point x =4and y =6isnot asolutionoftheequation2x 3y =7 because 2(4) 3(6) = 8 18 = 10, and 10 =7. 6 To get a geometric interpretation for what the set of solutions of 2x 3y =7 looks like, we can add 3y, subtract 7, and divide by 3 to rewrite 2x 3y =7 2 7 2 as 3x 3 = y. This is the equation of a line that has slope 3 and a y-intercept 7 of 3. In particular, the set of solutions to 2x 3y =7isastraightline. (This is why it’s called a linear equation.) 244 J\j -R 0 U’ oj -o
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1862 1862452 H 0 -o Systems of linear equations Rather than asking for the set of solutions of a single linear equation in two variables, we could take two di↵erent linear equations in two variables and ask for all those points that are solutions to both of the linear equations. For example, the point x =4andy =1isasolutionofbothoftheequations x + y =5andx y =3. If you have more than one linear equation, it’s called a system of linear equations, so that x + y =5 x y =3 is an example of a system of two linear equations in two variables. There are two equations, and each equation has the same two variables: x and y. A solution of a system of equations is a point that is a solution of each of the equations in the system.
Example. The point x =3andy =2isasolutionofthesystemoftwo linear equations in two variables 8x +7y =38 3x 5y = 1 because x =3andy =2isasolutionof3x 5y = 1 and it is a solution of 8x +7y =38. Unique solutions Geometrically, finding a solution of a system of two linear equations in two variables is the same problem as finding a point in R2 that lies on each of the straight lines corresponding to the two linear equations. Almost all of the time, two di↵erent lines will intersect in a single point, so in these cases, there will only be one point that is a solution to both equations. Such a point is called the unique solution of the system of linear equations.
Example. Let’s take a second look at the system of equations 8x +7y =38 3x 5y = 1 246 The first equation in this system, 8x +7y =38,correspondstoalinethat TheThe first first8 equation equation in in this this system, system, 8 8xx+7+7yy =38,correspondstoalinethat=38,correspondstoalinethat has slope 788. The second equation in this system, 3x 5y =3,isrepresented hashas slope slope 77.. The The second second equation equation3 3 in in this this system, system, 3 3xx 55yy =3,isrepresented=3,isrepresented by a line that has slope 3 = 3. Since the two slopes are not equal, the byby a a line line that that has has slope slope 53 == 53.. Since Since the the two two slopes slopes are are not not equal, equal, the the lines have to intersect in exactly 55 one55 point. That one point will be the unique lineslines have have to to intersect intersect in in exactly exactly one one point. point. That That one one point point will will be be the the unique unique solution. As we’ve seen before, x =3andy =2isasolutionofthissystem. solution.solution. As As we’ve we’ve seen seen before before that that xx =3and=3andyy =2isasolutiontothis=2isasolutiontothis It is the unique solution. system,system, it it must must be be the the unique unique solution. solution.
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248189 5 Exercises
1.) What are the coe cients of the equation 2x 5y = 23 ? 2.) What is the constant of the equation 2x 5y = 23 ? 3.) Is x = 4andy =3asolutionoftheequation2x 5y = 23 ?
4.) What are the coe cients of the equation 7x +6y =15? 5.) What is the constant of the equation 7x +6y =15? 6.) Is x =3andy = 10 a solution of the equation 7x +6y =15?
7.) Is x =1andy =0asolutionofthesystem x + y =1 2x +3y =3
8.) Is x = 1andy =3asolutionofthesystem 7x +2y = 1 5x 3 y = 14
9.) What’s the slope of the line 30x 6y =3? 10.) What’s the slope of the line 10x +5y =4? 11.) Is there a unique solution to the system 30x 6y =3 10x +5y =4
12.) What’s the slope of the line 6x +2y =4?
13.) What’s the slope of the line 15x +5y = 7? 249 14.) Is there a unique solution to the system 6x +2y =4 15x +5y = 7
For #15-17, find the roots of the given quadratic polynomials.
15.) 9x2 12x +4 16.) 2x2 3x +1 17.) 4x2 +2x 3
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